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saw activity at simple object and started a tiny section with examples.
Looking at this article: what is the need for having a zero object as opposed to just a terminal? In other words, would it be worse to say an object is simple if it has precisely two quotients, itself and $1$?
I’m particularly interested in the case of rings and commutative rings. One definition of simple ring is that it has precisely two two-sided ideals, and this condition is equivalent to the one I’m proposing. Similarly, a simple commutative ring under my proposal is just a field.
(It may be that the concept becomes interesting only under additional assumptions, such as Barr-exactness or something in that vein. That’s why I phrased my question as I did: would my proposal be any worse than the one given?)
The definition of simple ring on the nlab is $R$ is simple as a bimodule. This seems to be equivalent to your definition. As far as I can tell your definition should subsume that definition of simple ring. I think the terminology becomes problematic for Lie algebras however.
Yeah, for Lie algebras there are practical reasons for excluding the 1-dimensional case.
But I’m asking a more general theoretical question (insofar as there should be a general “theory” of simple objects in categories).
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